Reading: Chapter 4

In-class assignment, problem 2

Let's spend a little time working out problem 2 of the in-class assignment from Lecture 4.

Mersenne's Laws

During our slinky lab we noticed that stretching the slinky to twice its original length of 1 meter resulted in a doubling of wave speed while frequency stayed the same. What's that about? Marin Mersenne (1588-1648) determined three laws that help us answer this question. They focus primarily on frequency, but we already know the relationship between frequency, wavelength and wave speed.

First we might ask ourselves which variables (or fundamental descriptors) are important. Think of a violin. The strings have different thicknesses, and it is tuned by changing the tension on the strings.

• What affect does slinky mass have on this? This might be a good time to use linear density (or mass per length) as a fundamental descriptor.
• What about tension? Does it generally increase or decrease wave speed?
• Also, what affect does string length have on frequency, all other things being equal? What does doubling length do to frequency?

Table 3-4 of your textbook gives the three laws attributed to Mersenne. They say that:

	Doubling the string length (L) halves the frequency (f)
	Doubling the TENSION (F) results in an increase in frequency by square root of two
	Doubling the mass per length (W) results in reducing frequency by square root of two

All of these laws can be summarized as follows:

This explains why the strings of guitars, violas, etc., are of different thicknesses and kept at different tensions. In terms of Mersenne's equations, then, try to answer the following questions:

• Why are strings for lower open notes (think low E on a guitar) thicker than those for higher notes? What is 'wire-wound' about?
• Which string on a violin is under the highest tension and why?
• Why are the strings of a cello longer than those of a viola?
• How would doubling the mass per length of a string, without doing anything else, change the frequency? (be quantitative).

Some of you were wondering about how one gets different frequency standing waves (fundamental, 2nd harmonic, 3rd harmonic, etc.) on a string all at the same time. Here is another applet (courtesy of falstad.com) that let's you pluck a string (try it in different places) and see how this affects the mixture of standing waves (or overtone content) of the string.

Be sure and turn on sound so you can hear what plucking at different locations sounds like. Also, if you select "Display Modes" you can see and hear how the different overtones appear and decay over time, giving a guitar it's unique sound.

Here's the applet:

You guitar players may want to check out Ian Billington's web page on The Physics of the Acoustic Guitar.

Longitudinal Standing Waves (pipes)

Many musical instruments use methods other than the excitation of stretched strings to create musical sounds. In some instruments, sound waves within air are directly generated by exciting changes in the motion air molecules within and near the ends of the pipe.

These types of standing waves are longitudinal (in the direction of wave travel) as contrasted with transverse waves of a stretched string.

One can 'monitor' the air inside the pipe in two ways--

1. pressure (average spacing between air molecules, versus unaffected atmospheric pressure)
2. air velocity (net transport of air molecules, versus unaffected air velocity)

The relationship between these two is really not so apparent (at least to me!). Let's look at Figure 3-13 of your book and think about what it means:

At an air velocity antinode (marked "A" on the figures), air molecule 'layers' are rushing to the left or to the right with maximal variations in velocity over time. However, the 'layers' are about the same separation at those points (look under an "A" on each of the four (successive time) plots). So the air pressure there is constant

It turns out that a pressure node (a place where air pressure stays the same, like string displacement on the bridge of a guitar) is an air velocity antinode, and vice versa. In other words, an air pressure antinode-- a place where air pressure changes most-- is a place where air velocity doesn't change at all over time.

A few important TIPS about pipes

So, whenever you see a tube with sine waves drawn in it (like above), your first question should be:

"What does that graph represent, air velocity or pressure?"

The next step is to remember that:

"Air velocity nodes (N's) are equivalent to pressure antinodes (A's)-- and vice versa"

(The figures in your textbook typically show air velocity in plots of standing waves in pipes.)

Ends, nodes, antinodes and flips

Now it worthwhile to think about (boundary) conditions at the closed or open ends of a pipe. This will tell us where we expect to find nodes and antinodes. It will also help us understand if and where sound waves "flip" (reverse phase) when they reach an open or a closed end.

• What will the air velocity do at the closed end of the pipe? Will that be a node or antinode for air velocity?
• What about the open end? Will that be an air velocity node or antinode?
• What about pressure, where will the nodes and antinodes be?

The air velocity at the closed end must be zero. Air layers can't move to the left, into the closed end. So there is an air velocity node (N) at the closed end. Remember that this corresponds to an air pressure antinode. At the other end, the air pressure must be the same as that outside the pipe, so there is an air pressure node at the open end. This corresponds to an air velocity antinode (A).

What about wave pulses reflecting off of an open or closed end. What happens there? It's scientific process time!

• What do we think will happen to a pulse at an open end? Well, first what does it mean for an air pulse to 'flip' or not?
• I find it easier to think about pressure variations from normal (atmospheric) pressure. Why don't we define (arbitrarily) an upwards pulse as positive-- air pressure that becomes greater than normal at first. A negative pulse is air pressure that is initially less than normal. A 'flipped' pulse constitutes a change from initially greater than normal to initially less than normal.
• So what do we predict will happen upon reflection at an open end?
• Let's try a groovy demonstration to see.

  

  

  We find that sound pulses 'flip' upon reflecting off an open pipe end.

And they don't flip upon encountering a closed end.

I think about the open end of a pipe as a pressure node--where pressure is 'clamped' to outside (atmospheric) pressure. We noted in our slinky lab that (transverse) slinky pulses flipped upon reflecting off of 'clamped' ends. Ditto for sound waves encountering ('pressure clamped') open pipe ends.

Overtone Series for Pipes

Now that we know what to expect for open and closed pipe ends, we can work out the overtone series (set of standing waves that fit in...) for pipes.

Let's consider, first, a pipe with both ends open. We start by drawing a graph of air velocity (following the book's lead) for the simplest standing wave that 'fits' in the pipe.

So we remember that the open ends of pipes are air velocity antinodes (A's). And the simplest standing wave that fits is one with one air velocity node (N) in the middle.

To determine the wavelength (l) of this standing wave, compared the the length (L) of the pipe, we recognize that we must 'complete' the sine wave by adding to both ends. This gives us that:

l = 2 L

for the fundamental (or 1st harmonic) standing wave for an open-ended pipe. And we remember that:

and with that method, we can develop both a pictoral representation of the overtone series (Figure 3-20 of your textbook), and a table showing these and giving corresponding wavelengths and frequencies.

Table of nodes, etc., wavelengths and frequencies for open-ended pipe standing waves

Harmonic number (N)	# nodes	# antinodes	wavelength (l)	frequency (f=v/l)	AKA
1	1	2	2L	v/2L (call this f1)
2	2	3	L (2/2 L)	v/L (= 2 f1)
3	3	4	2/3 L	v/(2/3 L) (=3fl)
4	4	5	1/2 L (2/4 L)	v/(1/2 L) (=4fl)
5	5	6	2/5 L	v/(2/5 L) (=5fl)
n	n	n+1	2/n L	v / (2/n L) (=n fl)	(n is the number of nodes in this case)
(somewhat superflous comments follow)	not Modes	I personally have nothing against nodes	L is the length of the string	a wee high frequency	to stand or not to stand.. that is the question

We leave it to you to develop on your own graphs of (pictoral representations of) air velocity in a pipe with one end closed and the other open. This situation is representative of a flute, for example. You can see Figure 3-20 of your book and Table 3-7 to see if you understand how this works.

Synthesis of Complex Waves

To understand how one can synthesize the sounds of different musical instruments, it is useful to understand, in a conceptual way, something called Fourier synthesis.

Fourier synthesis, named after French mathematician Jean Baptiste Joseph Fournier, is a high-faluting way of saying

"let's add sine waves together and see what we get."

So let's just do that. We can use the beautiful Fourier Synthesis Applet, courtesy of Falstad.com, for this:

(Look for the applet here)

Using this applet we note that we can compose somewhat complex waveforms (triangular, square, sawtooth, etc.) using appropriate combinations (where we choose frequency, amplitude and phase... so many choices!) of pure sine waves.

What about going the other way around? Can we take a signal (say a "sawtooth" wave) and somehow determine which sine waves comprise it? If so, can we determine what relationships exist between the successive frequencies, their amplitudes and the phases of the Fourier components of that signal? Time to break out the demo room's trusty microphone, LabPro and LoggerPro software to determine the Fourier components of a sawtooth wave.

This process of analyzing seemingly complex signals in terms of sums of sine and cosine waves is, amazingly, called Fourier Analysis

This is not to be confused with Freudian Analysis.

Let's analyze various musical tones with Raven Software and see if some of this makes sense in that setting. Let's start with a single note played on a clarinet.

Our goal is to synthesize the sounds of musical instruments. So it is useful to combine some Fourier analysis with a little physics of sound and music to guide how we do that.

 

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